Classes of modules closed under transfinite extensions often provide for precovers, and hence fit in the machinery of relative homological algebra. However, there are important exceptions: the Whitehead groups, and flat Mittag-Leffler modules over non-perfect rings. The latter class is just the zero dimensional instance (for T = R and n = 0) of non-precovering of the class of all locally T-free modules, where T is any n-tilting module which is not Σ-pure split. The phenomenon occurs even for finite dimensional algebras, when R is hereditary of infinite representation type, and T is the Lukas tilting module. The key tools here are the tree modules, which have recently been generalized in order to solve Auslander’s problem on the existence of almost split sequences.
This video was produced by Syracuse University Department of Mathematics as part of ICRA 2016.
